Problem: Simplify the following expression: $\dfrac{27z^4}{54z^5}$ You can assume $z \neq 0$.
$ \dfrac{27z^4}{54z^5} = \dfrac{27}{54} \cdot \dfrac{z^4}{z^5} $ To simplify $\frac{27}{54}$ , find the greatest common factor (GCD) of $27$ and $54$ $27 = 3 \cdot 3 \cdot 3$ $54 = 2 \cdot 3 \cdot 3 \cdot 3$ $ \mbox{GCD}(27, 54) = 3 \cdot 3 \cdot 3 = 27 $ $ \dfrac{27}{54} \cdot \dfrac{z^4}{z^5} = \dfrac{27 \cdot 1}{27 \cdot 2} \cdot \dfrac{z^4}{z^5} $ $\phantom{ \dfrac{27}{54} \cdot \dfrac{4}{5}} = \dfrac{1}{2} \cdot \dfrac{z^4}{z^5} $ $ \dfrac{z^4}{z^5} = \dfrac{z \cdot z \cdot z \cdot z}{z \cdot z \cdot z \cdot z \cdot z} = \dfrac{1}{z} $ $ \dfrac{1}{2} \cdot \dfrac{1}{z} = \dfrac{1}{2z} $